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Magnetism

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Magnets and Magnetic Fields Electric Currents Produce Magnetic Fields Force on an Electric Current in a Magnetic Field; Definition of B Force on an Electric Charge Moving in a Magnetic Field Discovery and Properties of the Electron Mass Spectrometer

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Magnets have two ends – poles – called north and south. Like poles repel; unlike poles attract. Magnets and Magnetic Fields

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However, if you cut a magnet in half, you don’t get a north pole and a south pole – you get two smaller magnets. Magnets and Magnetic Fields

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Magnetic fields can be visualized using magnetic field lines, which are always closed loops. Magnets and Magnetic Fields

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Magnetic field lines: –Start from the north pole and end at the south pole. –Do not intersect –Tangent to the lines –Density of the lines

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The Earth’s magnetic field is similar to that of a bar magnet. Note that the Earth’s “North Pole” is really a south magnetic pole, as the north ends of magnets are attracted to it. Magnets and Magnetic Fields

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A uniform magnetic field is constant in magnitude and direction. The field between these two wide poles is nearly uniform. Magnets and Magnetic Fields

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Experiment shows that an electric current produces a magnetic field. The direction of the field is given by a right-hand rule. Electric Currents Produce Magnetic Fields

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Here we see the field due to a current loop; the direction is again given by a right-hand rule.

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A magnet exerts a force on a current- carrying wire. The direction of the force is given by a right- hand rule. Definition of the Magnetic Field

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The force on the wire depends on the current, the length of the wire, the magnetic field, and its orientation: This equation defines the magnetic field B. In vector notation: Definition of the Magnetic Field

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Unit of B : the tesla, T: 1 T = 1 N/A·m. Another unit sometimes used: the gauss ( G ): 1 G = T. Definition of the Magnetic Field

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Force on an Electric Current Magnetic Force on a current-carrying wire. A wire carrying a 30-A current has a length l = 12 cm between the pole faces of a magnet at an angle θ = 60°, as shown. The magnetic field is approximately uniform at 0.90 T. We ignore the field beyond the pole pieces. What is the magnitude of the force on the wire?

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Force on an Electric Current Measuring a magnetic field. A rectangular loop of wire hangs vertically as shown. A magnetic field B is directed horizontally, perpendicular to the wire, and points out of the page at all points. The magnetic field is very nearly uniform along the horizontal portion of wire ab (length l = 10.0 cm) which is near the center of the gap of a large magnet producing the field. The top portion of the wire loop is free of the field. The loop hangs from a balance which measures a downward magnetic force (in addition to the gravitational force) of F = 3.48 x N when the wire carries a current I = A. What is the magnitude of the magnetic field B?

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Force on an Electric Current Magnetic Force on a semicircular wire. A rigid wire, carrying a current I, consists of a semicircle of radius R and two straight portions as shown. The wire lies in a plane perpendicular to a uniform magnetic field B 0. Note choice of x and y axis. The straight portions each have length l within the field. Determine the net force on the wire due to the magnetic field B 0.

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The force on a moving charge is related to the force on a current: Once again, the direction is given by a right-hand rule. Force on an Electric Charge

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Negative charge near a magnet. A negative charge -Q is placed at rest near a magnet. Will the charge begin to move? Will it feel a force? What if the charge were positive, +Q ?

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Force on an Electric Charge Magnetic force on a proton. A magnetic field exerts a force of 8.0 x N toward the west on a proton moving vertically upward at a speed of 5.0 x 10 6 m/s (a). When moving horizontally in a northerly direction, the force on the proton is zero (b). Determine the magnitude and direction of the magnetic field in this region. (The charge on a proton is q = +e = 1.6 x C.)

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Force on an Electric Charge Magnetic force on ions during a nerve pulse. Estimate the magnetic force due to the Earth’s magnetic field on ions crossing a cell membrane during an action potential. Assume the speed of the ions is m/s.

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If a charged particle is moving perpendicular to a uniform magnetic field, its path will be a circle. Force on an Electric Charge

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Electron’s path in a uniform magnetic field. An electron travels at 2.0 x 10 7 m/s in a plane perpendicular to a uniform T magnetic field. Describe its path quantitatively.

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Force on an Electric Charge Stopping charged particles. Can a magnetic field be used to stop a single charged particle, as an electric field can?

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Force on an Electric Charge

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A helical path. What is the path of a charged particle in a uniform magnetic field if its velocity is not perpendicular to the magnetic field?

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Force on an Electric Charge The aurora borealis (northern lights) is caused by charged particles from the solar wind spiraling along the Earth’s magnetic field, and colliding with air molecules.

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Force on an Electric Charge Velocity selector, or filter: crossed E and B fields. Some electronic devices and experiments need a beam of charged particles all moving at nearly the same velocity. This can be achieved using both a uniform electric field and a uniform magnetic field, arranged so they are at right angles to each other. Particles of charge q pass through slit S 1 and enter the region where E points into the page and B points down from the positive plate toward the negative plate. If the particles enter with different velocities, show how this device “selects” a particular velocity, and determine what this velocity is.

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Electron Electrons were first observed in cathode ray tubes. These tubes had a very small amount of gas inside, and when a high voltage was applied to the cathode, some “cathode rays” appeared to travel from the cathode to the anode.

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Electron The value of e / m for the cathode rays was measured in 1897 using the apparatus below; it was then that the rays began to be called electrons. Figure goes here.

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Electron

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Millikan measured the electron charge directly shortly thereafter, using the oil-drop apparatus diagrammed below, and showed that the electron was a constituent of the atom (and not an atom itself, as its mass is far too small). The currently accepted values of the electron mass and charge are m = 9.1 x kg e = 1.6 x C

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A mass spectrometer measures the masses of atoms. If a charged particle is moving through perpendicular electric and magnetic fields, there is a particular speed at which it will not be deflected, which then allows the measurement of its mass: Mass Spectrometer

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All the atoms reaching the second magnetic field will have the same speed; their radius of curvature will depend on their mass. Mass Spectrometer

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Mass spectrometry. Carbon atoms of atomic mass 12.0 u are found to be mixed with another, unknown, element. In a mass spectrometer with fixed B ′, the carbon traverses a path of radius 22.4 cm and the unknown’s path has a 26.2-cm radius. What is the unknown element? Assume the ions of both elements have the same charge.

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Magnets have north and south poles. Like poles repel, unlike attract. Unit of magnetic field: tesla. Electric currents produce magnetic fields. A magnetic field exerts a force on an electric current: Summary

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A magnetic field exerts a force on a moving charge: Summary

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Magnetic Field

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Magnetic Field Due to a Straight Wire Force between Two Parallel Wires Definitions of the Ampere and the Coulomb Ampère’s Law Magnetic Field of a Solenoid and a Toroid Electromagnets and Solenoids – Applications

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The magnetic field due to a straight wire is inversely proportional to the distance from the wire: The constant μ 0 is called the permeability of free space, and has the value μ 0 = 4π x T·m/A. Field Due to a Straight Wire

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Calculation of B near a wire. An electric wire in the wall of a building carries a dc current of 25 A vertically upward. What is the magnetic field due to this current at a point P 10 cm due north of the wire?

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Field Due to a Straight Wire Magnetic field midway between two currents. Two parallel straight wires 10.0 cm apart carry currents in opposite directions. Current I 1 = 5.0 A is out of the page, and I 2 = 7.0 A is into the page. Determine the magnitude and direction of the magnetic field halfway between the two wires.

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Field Due to a Straight Wire Magnetic field due to four wires. This figure shows four long parallel wires which carry equal currents into or out of the page. In which configuration, (a) or (b), is the magnetic field greater at the center of the square?

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The magnetic field produced at the position of wire 2 due to the current in wire 1 is The force this field exerts on a length l 2 of wire 2 is Force between Parallel Wires

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Parallel currents attract; antiparallel currents repel. Force between Parallel Wires

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Force between two current-carrying wires. The two wires of a 2.0-m-long appliance cord are 3.0 mm apart and carry a current of 8.0 A dc. Calculate the force one wire exerts on the other.

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Force between Parallel Wires Suspending a wire with a current. A horizontal wire carries a current I 1 = 80 A dc. A second parallel wire 20 cm below it must carry how much current I 2 so that it doesn’t fall due to gravity? The lower wire has a mass of 0.12 g per meter of length.

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Definitions of the Ampere and the Coulomb The ampere is officially defined in terms of the force between two current-carrying wires: One ampere is defined as that current flowing in each of two long parallel wires 1 m apart, which results in a force of exactly 2 x N per meter of length of each wire. The coulomb is then defined as exactly one ampere-second.

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Biot-Savart Law The Biot-Savart law gives the magnetic field due to an infinitesimal length of current; the total field can then be found by integrating over the total length of all currents:

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Biot-Savart Law B due to current I in straight wire. For the field near a long straight wire carrying a current I, show that the Biot-Savart law gives B = μ 0 I/2πr.

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Solution:

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Biot-Savart Law Current loop. Determine B for points on the axis of a circular loop of wire of radius R carrying a current I.

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Solution: The perpendicular components of B cancel, leaving along only the component the axis.

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Biot-Savart Law B due to a wire segment. One quarter of a circular loop of wire carries a current I. The current I enters and leaves on straight segments of wire, as shown; the straight wires are along the radial direction from the center C of the circular portion. Find the magnetic field at point C.

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Solution: The straight segments of wire produce no field at C, as the current is parallel to r.

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Ampère’s law relates the magnetic field around a closed loop to the total current flowing through the loop: Ampère’s Law This integral is taken around the edge of the closed loop.

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Ampère’s Law Using Ampère’s law to find the field around a long straight wire: Use a circular path with the wire at the center; then B is tangent to dl at every point. The integral then gives so B = μ 0 I/2πr, as before.

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Ampère’s Law Field inside and outside a wire. A long straight cylindrical wire conductor of radius R carries a current I of uniform current density in the conductor. Determine the magnetic field due to this current at (a) points outside the conductor ( r > R ) and (b) points inside the conductor ( r < R ). Assume that r, the radial distance from the axis, is much less than the length of the wire. (c) If R = 2.0 mm and I = 60 A, what is B at r = 1.0 mm, r = 2.0 mm, and r = 3.0 mm?

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Ampère’s Law Coaxial cable. A coaxial cable is a single wire surrounded by a cylindrical metallic braid. The two conductors are separated by an insulator. The central wire carries current to the other end of the cable, and the outer braid carries the return current and is usually considered ground. Describe the magnetic field (a) in the space between the conductors, and (b) outside the cable.

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Ampère’s Law A nice use for Ampère’s law. Use Ampère’s law to show that in any region of space where there are no currents the magnetic field cannot be both unidirectional and nonuniform as shown in the figure.

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Ampère’s Law Solving problems using Ampère’s law: Ampère’s law is only useful for solving problems when there is a great deal of symmetry. Identify the symmetry. Choose an integration path that reflects the symmetry (typically, the path is along lines where the field is constant and perpendicular to the field where it is changing). Use the symmetry to determine the direction of the field. Determine the enclosed current.

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Magnetic Field of a Solenoid and a Toroid A solenoid is a coil of wire containing many loops. To find the field inside, we use Ampère’s law along the path indicated in the figure.

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Solenoid and Toroid The field is zero outside the solenoid, and the path integral is zero along the vertical lines, so the field is ( n is the number of loops per unit length)

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Solenoid and Toroid Field inside a solenoid. A thin 10-cm-long solenoid used for fast electromechanical switching has a total of 400 turns of wire and carries a current of 2.0 A. Calculate the field inside near the center.

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Solenoid and Toroid Toroid. Use Ampère’s law to determine the magnetic field (a) inside and (b) outside a toroid, which is like a solenoid bent into the shape of a circle as shown.

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Remember that a solenoid is a long coil of wire. If it is tightly wrapped, the magnetic field in its interior is almost uniform. Electromagnets and Solenoids – Applications

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If a piece of iron is inserted in the solenoid, the magnetic field greatly increases. Such electromagnets have many practical applications. Electromagnets and Solenoids – Applications

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Magnitude of the field of a long, straight current-carrying wire: The force of one current-carrying wire on another defines the ampere. Ampère’s law: Summary

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Magnetic field inside a solenoid: Biot-Savart law: Summary Ferromagnetic materials can be made into strong permanent magnets.

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